no, the center is the set of all scalar matrices, ie, A is in the center iff A = c*I for some c in F\{0}

consider elementary matrices

another suggestion, when working on problems like this where you don't know what it is you want to prove, think simple, ie, take n = 2, and A = (1 0; 0 -1) which is diagonal, and B = (2 3; 1 1), then AB != BA, so A is not in the center but A is a diagonal matrix.